Accounting for Future Redesign in the Optimization …...redesign small is obvious in that redesign can lead to project delays and increase costs. On the other hand, a reduction in

Accounting for Future Redesign in the Optimization of

an Integrated Thermal Protection System

Diane Villanueva1,2∗, Raphael T. Haftka1†, and Bhavani V. Sankar1 ‡

1University of Florida, Gainesville, FL, 32611, USA2Ecole Nationale Superieure des Mines de Saint-Etienne, Saint-Etienne, France

Most components undergo tests after they are designed and are redesigned if necessary.Tests help designers find unsafe and overly conservative designs, and redesign can restoresafety or reduce mass. In general, the changes to the mass and reliability of the designafter the test and redesign are not considered. This paper examines the effect of controllingdesign parameters and the parameters that control the test and redesign when optimizingthe design on an integrated thermal protection system of a space vehicle. This studyexplores the tradeoff between the probability of redesign and the mass at the design stage.We observed that to minimize mass, designs should initially satisfy safety requirementssuch that most redesigns concern overly conservative designs. The additional knowledgegained from the test allows the designer to consider a smaller safety margin for the redesignto reduce the mass.

Nomenclature

d = design variable

ec = computational error

ex = experimental error

f(T ) = probability distribution of the temperature

m = mass per unit area, kgm2

pf = probability of failure, %

r = random variable

S = safety margin

∆T = change in temperature, K

T = temperature, K

subscripts

0 = initial

calc = calculated

corr = corrected

nom = nominal

meas = measured

test = test article

true = true

∗Graduate Research Assistant, Mechanical and Aerospace Engineering/Institut Henri Fayol, AIAA student member†Distinguished Professor, Mechanical and Aerospace Engineering, AIAA Fellow‡Ebaugh Professor, Mechanical and Aerospace Engineering, AIAA Associate Fellow

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American Institute of Aeronautics and Astronautics

I. Introduction

In reliability based design optimization, uncertainties are considered when calculating the reliability ofthe structure. In the design process, uncertainty is often compensated for with safety factors and knockdownfactors. However, after design, it is customary for the component to undergo various uncertainty reductionmeasures (URMs). Examples of URMs in the aerospace field include thermal and structural testing, inspec-tion, health monitoring, maintenance, and improved analysis and failure modeling. Since most componentsundergo these URMs, it would be beneficial to include their effects in the design process.

In recent years, there has been a movement to quantify the effect of URMs on the safety of the productover its life cycle. Much work has been completed in the areas of inspection and maintenance for structuresunder fatigue loading.1–4 Studies by Acar et al.5 investigated the effects of future tests and redesign onthe final distribution of failure stress and structural design with varying numbers of tests at the coupon,element, and certification levels. Sankararaman et al.6 proposed an optimization algorithm of test resourceallocation for multi-level and coupled systems.

Previously, Villanueva et al.7 proposed a method to incorporate the effect of a single future thermal testfollowed by redesign on the initial reliability estimates of an integrated thermal protection system (ITPS). AnITPS is a structure on a reusable launch vehicle that simultaneously provides protection from aerodynamicheating during reentry, while working as a load bearing structure. Monte Carlo sampling of the assumedcomputational and experimental errors was used to sample future test alternatives, or the possible outcomesof the future test. Using the future alternatives, the methodology included two methods of calibration andredesign, needed when the test result indicates that the design is unacceptable. It was observed that thedeterministic approach to calibration and redesign, which acts to restore the original (designed) safety margin,leads to a greatly reduced probability of failure after the test and redesign, a reduction that usually is notquantified. A probabilistic approach was also presented, which provided a way to more accurately estimatethe probability of failure after the test and allowed the tradeoff of weight against performing additional tests.Matsumura et al.8 extended the methodology to include additional failure modes of the ITPS.

This paper examines the effect of controlling design parameters and the parameters that control the futuretest and redesign when optimizing the design of the ITPS. The focus is on tuning the redesign conditions(i.e., the rules that dictate whether or not redesign is performed) along with the design such that boththe mass and probability of an unacceptable test result are minimized. The desire to keep probability ofredesign small is obvious in that redesign can lead to project delays and increase costs. On the other hand,a reduction in mass is attractive, especially for space vehicles which have extremely large per-weight launchcosts. This study will explore the tradeoff between the probability of redesign and the mass at the designstage.

A. Integrated Thermal Protection Shield Description

Figure 1. Corrugated core sandwich panel ITPS con-cept

Figure 1 shows the ITPS panel that is studied, whichis a corrugated core sandwich panel concept. Thedesign consists of a top face sheet and webs madeof titanium alloy (Ti-6Al-4V), and a bottom facesheet made of beryllium. Saffil R© foam is used asinsulation between the webs. The relevant geometricvariables of the ITPS design are also shown on theunit cell in Figure 1. These variables are the top facethickness (tT ), bottom face thickness (tB), thicknessof the foam (dS), web thickness (tw), corrugationangle (θ), and length of unit cell (2p). The mass perunit area is calculated using Eq.(1)

m = ρT tT + ρBtB +ρwtwdSp sin θ

(1)

where ρT , ρB , and ρw are the densities of the materials that make up the top face sheet, bottom face sheet,and web, respectively. Additional information on the integrated thermal protection system is provided inthe Appendix.

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In this study, we consider thermal failure to occur when the temperature of the bottom face sheetexceeds an allowable temperature. We assume that tests of the structure will be conducted to verify thedesign. Observed data from the test will be utilized to calibrate errors in analytical calculations.

II. Analysis and Post-Design Test with Redesign

It is assumed that an analyst has a computational model by which to calculate the change in the temper-ature of the bottom face sheet of the ITPS, ∆Tcalc, for a design described by design variables d and randomvariables r. The randomness is due to variabilities in material properties, manufacturing, and environmentaleffects. Using ∆Tcalc, the calculated temperature is defined as

Tcalc(d, r, v0) = T0(1 − v0) + ∆Tcalc(d, r) (2)

where T0 is the initial temperature of the bottom face sheet, which also has variability represented by v0.The design is obtained via a deterministic optimization problem which requires that the calculated

temperature be less than or equal to some deterministic allowable temperature T detallow by some a safety

margin S1 as shown in Eq.(3).

mind={tw,tB ,dS}

m(d)

subject to T0 + ∆Tcalc(d, rnom) + S1 ≤ T detallow

tw,l ≤ tw ≤ tw,u

tB,l ≤ tB ≤ tB,u

dS,l ≤ dS ≤ dS,u

(3)

Note that for the deterministic design, the random variables are held at the nominal value rnom and thevariability in the initial temperature is zero. The subscripts l and u on the design variables represent thelower and upper bounds, respectively.

After the design stage, a test is conducted to verify the chosen design. The test is performed on a testarticle described by dtest and rtest

a and an experimentally measured change in temperature, ∆Tmeas, isfound. For this test design, ∆Tcalc(dtest, rtest) and Tcalc(dtest, rtest) are also calculated.

As a means of calibration, the experimentally measured and calculated temperatures can be used in theform of a correction factor θ for the computational model. That is, the corrected calculated temperature isgiven as

Tcalc,corr(d, r, v0) = T0(1 − v0) + θ∆Tcalc(d, r)

where θ =∆Tmeas

∆Tcalc(dtest,rtest)

(4)

Note that this results in an updated distribution of the corrected-calculated temperature.Should the test result show that a design is unacceptable, redesign occurs. The criterion for redesign is

based on the safety margin of the corrected calculated temperature of the original design. The lower andupper limits of the safety margin of the corrected temperature are represented with S2 and S3, respectively.This is expressed as

Redesign if: T detallow − (T0 + θ∆Tcalc(d, rnom)) ≤ S2

or T detallow − (T0 + θ∆Tcalc(d, rnom)) ≥ S3

(5)

Deterministic redesign is performed so that the corrected calculated temperature of the redesign (with thecorrection factor) is less than or equal to the allowable temperature by a safety margin S4. This safetymargin S4 does not necessarily need to be equal to the initial safety margin S1. For example, since moreinformation is gained from the test, the designer may choose to design to save weight by reducing the safetymargin. This can formulated into an optimization problem to minimize the mass given a constraint on the

aIt is assumed that the test article design is accurately measured such that rtest is deterministic and there is no variabilityin the initial temperature.

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corrected calculated temperature of the new redesign, where the design variables are the geometry.

mind={tw,tB ,dS}

m(d)

subject to T0 + θ∆Tcalc(d, rnom)redesign + S4 ≤ T detallow

tw,l ≤ tw ≤ tw,u

tB,l ≤ tB ≤ tB,u

dS,l ≤ dS ≤ dS,u

(6)

III. Uncertainty Definition

Oberkampf et al.9 provided an analysis of different sources of uncertainty in engineering modeling andsimulation, which was simplified by Acar et al.10 We use classification similar to Acar’s to categorize typesof uncertainty as errors (uncertainties that apply equally to every ITPS) or variability (uncertainties thatvary in each individual ITPS). We further describe errors as epistemic and variability as aleatory.

Variability is modeled as random uncertainties that can be modeled probabilistically. We simulate thevariability through a Monte Carlo simulation (MCS) that generates values of the random variables r basedon an estimated distribution and calculates the change bottom face sheet temperature ∆Tcalc. In addition,we sample the variability in the initial temperature v0. This forms the temperature Tcalc for each sample,generating the probability distribution function. The calculated temperature distribution that reflects therandom variability is denoted fcalc(T ). Additionally, we have variability in the allowable temperature Tallow.

In contrast to variability, errors are fixed for a given ITPS and the true values are largely unknown, sothey can be modeled probabilistically as well. We have classified two sources of error, which are describedin Table 1.

Table 1. Description of Errors

Symbol Description

ec computational error due to modeling of the temperature change ∆Tcalc

ex experimental error in measuring ∆Tmeas

In estimating the temperature of a design, the error must also be considered. As previously described,the calculated temperature distribution fcalc(T ) of the design reflects random variability. If the true value ofthe computational error is known, then the true temperature distribution, ftrue(T ), associated with fcalc(T )is known, as shown in Fig. 2(a). The true temperature still has randomness due to the variabilities.

(a) Calculated, true, and allowable temperature distri-butions

(b) Calculated, allowable, and sampled possible truetemperature distributions

Figure 2. Example illustrating (a) known calculated and allowable temperature distributions and unknowntrue distribution, (b) sampling to obtain possible true temperature distributions

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Since the error is unknown and modeled probabilistically, we instead sample the computational errorto create several possible distributions of the true temperature distributions,f iP true(T ) corresponding to theith sample of ec. This sampling is illustrated in Fig. 2(b).Using the allowable temperature distribution, theprobability of failure can be calculated for each sample of the computational error. This forms a distributionof the probability of failure.

IV. Distribution of the Probability of Failure

The true temperature for a design described by geometric design variables d and random variables r canbe defined as

Ttrue(d, r, v0) = T0(1 − v0) + (1 − ec,true)∆Tcalc(d, r) (7)

The limit state for the probability of failure takes into account the variability in the allowable tempera-tureb along with the distribution of the true temperature. The limit state equation g is formulated as thedifference between a capacity C and response R as shown in Eq.(8).

gtrue = Tallow − Ttrue(d, r, v0) = C −R (8)

Using the limit state equation, the probability of failure is calculated using Separable Monte Carlo.11

The probability of failure pf is calculated with Eq.(9), where M and N are the number of capacity andresponse samples, respectively. The indicator function I is 1 if the g is less than zero and 0 otherwise.

pf =1

MN

N∑i=1

M∑j=1

I[gtrue(Cj , Ri) < 0] (9)

As described in the previous section, a distribution of the probability of failure can be formed by samplingthe computational error for ec,true and calculating the probability of failure for each sample.

V. Simulating Future Processes at the Design Stage

Monte Carlo sampling of the true values of the errors from the assumed distributions is used to simulatethe future test and redesign alternatives for a candidate design. The steps to simulate a single alternativeof the future test with possible redesign are listed below:

1. Sample the true errors from assumed distributions

2. Construct the virtual test article and use the true ec and ex samples to simulate a test result andcorrection factor θ

3. Apply the correction factor based on the test result to ∆Tcalc

4. Evaluate if redesign is necessary based on S2 and S3, then redesign, if necessary

5. Evaluate mass and probability of failure using surrogates

To simulate another alternative, the true errors are re-sampled and the process is repeated. Therefore,for n possible outcomes of a future test, we sample n sets of the errors, and obtain n true probabilities offailure and up to n updated designs. Note that there is a single initial design, but if k of the n cases arere-designed we will end up with up to k + 1 different designs.

VI. Optimization of the Geometry and the Redesign Procedure

A. Problem Description

We explored the effect of the future test and redesign when optimizing mass with the safety margins S1, S2,S3, and S4 as design variables. This was done through an optimization problem that minimizes the mean

bThe absence of the superscript “det” for Tallow denotes the allowable temperature with variability to distinguish it fromthe deterministic allowable temperature T det

allow.

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mass µm and probability of redesign pre subject to constraints on the mean probability of failure µpf, the

95th percentile of the probability of failure P95(pf ), the ranges of the safety margins, and the ranges of thedesign variables. The formulation is given in Eq.(10).

minS1,S2,S3,S4

µm, pre

subject to µpf≤ 0.1%

P95(pf ) ≤ 0.5%

35 ≤ Si ≤ 65 for i = 1, 4

S1 − 35 ≤ S2 ≤ S1

S1 ≤ S3 ≤ S1 + 35

1.24mm ≤ tw ≤ 1.77mm

4.94mm ≤ tB ≤ 7.06mm

49.9mm ≤ dS ≤ 71.3mm

(10)

The constraints on S1 and S4 restrict the two values to be within the window of 35 to 65 K, and they are notconstrained to have equal values. The constraints on S2 and S3 restrict the acceptable values of the safetymargin after correction to within 35 K of S1.

Table 2. Bounds of computational and experimentalerrors

Error Distribution Bounds

ec Uniform ±0.12

ex Uniform ±0.03

For this problem, the computational and experi-mental errors were distributed as described in Table2. Given these distributions, the correction factorθ ranged from 0.85 to 1.15. The distributions ofthe variables with uncertainty due to variability areprovided in the Appendix.

To reduce the computational cost of simulatinga future test, surrogates of the mass and reliabil-ity index were developed. The reliability index β is

related to the probability of failure by pf = Φ(−β), where Φ is the standard normal cumulative densityfunction. For example, for a probability of failure of 0.1%, the reliability index is 3.72. The development ofthese surrogates is described in the Appendix.

The problem in Eq.(10) was solved by forming a cloud of 10,000 points using Latin Hypercube samplingof the design variables S1, S2, S3, and S4. For each set of design variables, 10,000 future alternatives weresampled to obtain the distributions of the mass and probability of failure, and the probability of redesign.The probability of redesign was calculated by dividing the number of redesigns by the total number ofalternatives (i.e., 10,000). The set of points that satisfied the constraints on the probability of failure wasfound, and, from this set of feasible points, we formed the Pareto front for minimum probability of redesignand mean mass after redesign.

B. Results

As a point of comparison, we also found the Pareto front for minimum mass and mean probability of failurewithout redesign (assuming no tests are performed and no redesign). These were obtained for values of S1

ranging from 35 to 65 K. The Pareto front is shown in Fig. 3. The points on the Pareto front are describedin the Appendix. The design with the minimum mass of 24.70 kg/m2 was found with a safety margin S1 of48.9 K. The mean probability of failure was 0.098% and the 95th percentile was 0.5%.

The Pareto front for minimum probability of redesign and mean mass after redesign is displayed in Fig.4. The points on this Pareto front are also detailed in the Appendix. We observed that the majority ofinfeasible design have small mean mass values. This is obvious as lighter designs will have less insulation,and thus higher temperatures and larger probabilities of failure. Reductions in mean mass after redesignwere observed for probabilities of redesign greater than 10%. The mean mass values after redesign at thesepoints were less than the minimum mass of 24.70 kg/m2 obtained when redesign was not allowed.

Figure 5 shows the percent reduction in mass and probability of failure after redesign for the Paretofront. While the reductions in mean mass were less than 5% and the median mass remained mostly thesame, we observed large reductions in the mean and 95th percentile of the probability of failure. However,the increase in median probability of failure was large, particularly at large probabilities of redesign.

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Figure 3. Pareto front for minimum mass and mean probability of failure without redesign

Figure 4. Pareto front for minimum probability of redesign and mean mass after redesign. Infeasible andinfeasible points in the design space are shown, along with the before-redesign mass of the points on thePareto front

The values of the safety margins for the designs on the Pareto front are displayed in Fig. 6(a). We observedthat the initial safety margin S1 had already reduced from approximately 65 K to 49 K by pre = 5%. Thelower bound of the acceptable safety margin with correction S2 is at its lower bound, with a maximum valueof approximately 30 K at probabilities of redesign less than 5%. At probabilities of redesign greater than5%, S2 remained between 15 to 20 K, for which the difference from S1 is near the upper bound of 35 K. Thisresulted in the minimum probability of redesign of unconservative designs. In Fig. 6(b), which shows thepercentage of the total probability of redesign that is conservative and unconservative, we observed that thiswas indeed the case, and that less than 5% of the total probability of redesign was attributed to probabilityof unconservative redesign for all points on the Pareto front.

For the upper bound on acceptable safety margin with correction S3, we observed that the values werelarge (around 100 K) but gradually reduced to values near S1 at 49 K. This led to the gradual increase in

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Figure 5. Percent reduction in mass and probability of failure from before to after redesign. Note that anegative reduction indicates an increase in value from before to after redesign.

(a) Safety margin vs probability of redesign (b) Percentage of conservative and unconservative redesigns

Figure 6. For Pareto front for minimum probability of redesign and mean mass after redesign, (a) safety marginsversus total probability of redesign and (b) percentage of total probability of redesign that are unconservativeand conservative

probability of conservative redesign as the probability of unconservative redesign remained at low values.Thus, the probability of conservative redesign comprised the majority of the total probability of redesign forthe designs on the Pareto front. At the same time, we observed that the safety margin S4 of the redesignwas set to values below S1 and at values less than the minimum value without tests and redesign of 48.9 K.That is, after the test, the redesign has a smaller safety margin than possible for the original design. Thecombined effect of redesigning conservative designs for a reduced safety margin was a reduction in the meanmass and mean probability of failure with an increase in the median probability of failure.

VII. Summary and Conclusions

In this paper, we investigated the effect of including a future test and redesign on the optimum design ofthe integrated thermal protection system by tuning the variables that defined the initial design, the variablesthat control if redesign is performed, and and the variables that define the redesign. We examined points

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on a Pareto front for minimum probability of redesign and mean mass after redesign. At small values ofprobability of redesign, we initially observed that the design was slightly safer and heavier than the minimummass design without redesign under the same constraints on safety. As the probability of redesign increased,the initial designs tended toward the minimum mass design that would be possible if redesign was not allowed.As a larger probability of redesign was accepted, the redesign criterion dictated that mostly conservativedesigns undergo redesign, which resulted in a reduction in the mean and 95th percentile of probability offailure and an increase in the median probability of failure. At the same time, if redesign was performed,the safety margin of the new design was less than the original, which led to a reduction in mass.

The need to keep the probability of redesign small is obvious since redesign can cause project delaysand, consequently, increase costs. This study supports the idea of creating initial designs that satisfy theconstraints on safety while only redesigning to reduce cost, which for the ITPS, results from reduced mass.

Appendix

Integrated Thermal Protection System

Thermal analysis of the integrated thermal protection system is done using 1-D heat transfer equations on amodel of the unit cell. The heat flux incident on the top face sheet of the panel is highly dependent on thevehicle shape as well as the vehicle’s trajectory. As in previous studies by Bapanapalli,12 incident heat fluxon a Space Shuttle-like vehicle was used. A large portion of the heat is radiated out to the ambient by thetop face sheet, and the remaining portion is conducted into the ITPS. We consider the worst-case scenariowhere the bottom face sheet cannot dissipate heat by assuming the bottom face sheet is perfectly insulated.Also, there is no lateral heat flow out of the unit cell, so that heat flux on the unit cell is absorbed by thatunit cell only. For a more in-depth description of the model and boundary conditions, the reader is referredto the Bapanapalli reference.

The random variables used in this paper are described in Table 3.

Table 3. ITPS Random Variables

Variable Symbol Distribution Nominal Bounds/ CV(%) Correlation

web thickness tw Uniform - ±3

bottom face sheet tB Uniform - ±3

foam thickness ds Uniform - ±3

top face sheet thickness tT Uniform 1.2 mm ±3

half unit cell length p Uniform 34.1 mm ±3

angle of corrugation θ Uniform 80◦ ±3

density of titanium ρTi Normal 4429 kgm3 2.89 correlation

thermal conductivity of titanium kTi Normal 7.6 Wm/K 2.89 coefficient = 0.95

specific heat of titanium cTi Normal 564 Jkg/K 2.89

density of beryllium ρBe Normal 1850 kgm3 2.89 correlation

thermal conductivity of beryllium kBe Normal 203 Wm/K 3.66 coefficient = 0.95

specific heat of beryllium cBe Normal 1875 Jkg/K 2.89

density of foam ρS Normal 24 kgm3 5.78 correlation

thermal conductivity of foam kS Normal 0.105 Wm/K 5.78 coefficient = 0.95

specific heat of foam cS Normal 1120 Jkg/K 2.89

initial temperature vi Uniform 0 1

allowable temperature Tallow Lognormal 660 K 2.42 (T detallow = 623.15K)

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Surrogates for Mass and Probability of Failure

In this section, it is shown that the mass before and after redesign can be found using a surrogate that is afunction of safety margin and difference between the allowable temperature Tallow and initial temperature T0.A surrogate of of the probability of failure that is a function of the same two variables and the computationalerror ec can be made as well.

As shown in Eq.(3), the initial design satisfies

T0 + ∆Tcalc(d, r) + S1 = Tallow (11)

Rearranged so that ∆Tcalc(d, r) is on the left hand side, this becomes

∆Tcalc(d, r) = (Tallow − T0) − S1 (12)

By Eq.(6)the redesign should satisfy

T0 + θ∆Tcalc(d, r) + S4 = Tallow (13)

which rearranged so that ∆Tcalc(d, r) is on the left hand side is

∆Tcalc(d, r) = (Tallow − T0)/θ − S4/θ (14)

By Eqs.(12) and (14), the two are equivalent if (Tallow−T0) = [(Tallow−T0)/θ]afterredesign and S1 = S4/θ.Therefore, ∆Tcalc, along with its corresponding mass and probability of failure, is a function of (Tallow −T0)and S, where the values with and without redesign are related through θ. This allows the mass to becalculated simply using surrogates with the inputs (Tallow−T0) and S. A surrogate to obtain the probabilityof failure can also be obtained by including the computational error ec as an input.

Note that ∆Tcalc(d, r) does not need to be calculated because, for a given (Tallow − T0) and S1, we canfind ∆Tcalc(d, r) by

(Tallow − T0) − S1 = ∆Tcalc(d, r) (15)

When the correction is applied, then we evaluate if redesign is necessary by

Redesign if: (Tallow − T0) − θ[(Tallow − T0) − S1] ≤ S2

or (Tallow − T0) − θ[(Tallow − T0) − S1] ≥ S3

(16)

which simplifies toRedesign if: (Tallow − T0)(1 − θ) + θS1 ≤ S2

or (Tallow − T0)(1 − θ) + θS1 ≥ S3

(17)

Detailed Results

Pareto front for minimum mass and probability of failure without redesign

The Pareto front for minimum probability of failure and mass without redesign is described in detail in Table4.

Pareto front for minimum probability of redesign and mean mass after redesign

The Pareto front for minimum probability of redesign and mean mass after redesign is described in detail inTable 5.

Acknowledgments

The material is based upon work supported by NASA under award No.NNX08AB40A. Any opinions,findings, and conclusions or recommendations expressed in this material are those of the author(s) and donot necessarily reflect the views of the National Aeronautics and Space Administration.

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Table 4. Pareto front for minimum mass and mean probability of failure without redesign

S1 (K) m (kg/m2) µpf(%) P95(pf ) (%) Median pf (%)

57.7 25.08 0.022 0.117 2.05E-05

57.1 25.05 0.024 0.129 2.58E-05

52.6 24.85 0.054 0.280 1.71E-04

52.0 24.83 0.060 0.308 2.16E-04

51.2 24.79 0.068 0.350 2.98E-04

50.8 24.78 0.073 0.376 3.54E-04

49.6 24.73 0.088 0.450 5.55E-04

49.5 24.72 0.089 0.455 5.70E-04

49.2 24.71 0.095 0.484 6.64E-04

49.1 24.70 0.096 0.490 6.86E-04

49.0 24.70 0.098 0.499 7.16E-04

48.9 24.70 0.098 0.500 7.19E-04

48.9 24.70 0.098 0.500 7.19E-04

References

1Fujimoto, Y., C., K. S., Hamada, K., and Huang, F., “Inspection Planning Using Genetic Algorithm for Fatigue Dete-riorating Structures,” International Offshore and Polar Engineering Conference, Vol.4 , International Society of Offshore andPolar Engineers, Golden, CO, 1998, pp. 99–109.

2Toyoda-Makino, M., “Cost-based Optimal History Dependent Strategy for Random Fatigue Cracks Growth,” ProbabilisticEngineering Mechanics, Vol. 14, No. 4, October 1999, pp. 339–347.

3Garbatov, Y. and Soares, C. G., “Cost and Reliability Based Strategies for Fatigue Maintenance Planning of FloatingStructures,” Reliability Engineering and Systems Safety, Vol. 73, No. 3, 2001, pp. 293–301.

4Kale, A., Haftka, R. T., and Sankar, B. V., “Efficient Reliability-Based Design and Inspection of Panels Against Fatigue,”Journal of Aircraft , Vol. 45, No. 1, 2008, pp. 86–96.

5Acar, E., Haftka, R. T., and Kim, N. H., “Effects of Structural Tests on Aircraft Safety,” AIAA Journal , Vol. 48, No. 10,2010.

6Sankararaman, S., McLemore, K., Liang, C., Bradford, S. C., and Peterson, L., “Test resource allocation for uncertaintyquantification of multi-level and coupled systems,” 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,and Materials Conference, Denver, CO, 2011.

7Villanueva, D., Haftka, R. T., and Sankar, B. V., “Including the Effect of a Future Test and Redesign in ReliabilityCalculations,” AIAA Journal , Vol. 49, No. 12, 2011, pp. 2760–2769.

8Matsumura, T., Haftka, R. T., and Sankar, B. V., “Reliability Estimation Including Redesign Following Future Test foran Integrated Thermal Protection System,” 9th World Congress on Structural and Multidisciplinary Optimization, Shizuoka,Japan, 2011.

9Oberkampf, W. L., Deland, S. M., Rutherford, B. M., Diegert, K. V., and Alvin, K. F., “Error and Uncertainty inModeling and Simulation,” Reliability Engineering and System Safety, Vol. 75, 2002, pp. 333–357.

10Acar, E., Haftka, R. T., Kim, N. H., and Buchi, D., “Effects of structural tests on aircraft safety,” 50thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, May 2009.

11Smarslok, B. P., Haftka, R. T., Carraro, L., and Ginsbourger, D., “Improving accuracy of failure probability estimateswith separable Monte Carlo,” International Journal of Reliability and Safety, Vol. 4, 2010, pp. 393–414.

12Bapanapalli, S. K., Design of an Integrated Thermal Protection System for Future Space Vehicles, Ph.D. thesis, Universityof Florida, 2007.

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Table

5.

Pareto

frontform

inim

um

probability

ofredesign

and

mean

mass

afterredesign.

Des

ign

Var

iab

les

Pro

bab

ilit

yof

Red

esig

nB

efore

Red

esig

nA

fter

Red

esig

n

S1

S2

S3

S4

tota

lco

nu

nco

nm

µpf

P95(p

f)

med

pf

µm

P95(m

)m

edm

µpf

P95(p

f)

med

pf

(K)

(K)

(K)

(K)

(%)

(%)

(%)

(kg/m

2)

(%)

(%)

(%)

(kg/m

2)

(kg/m

2)

(kg/m

2)

(%)

(%)

(%)

64.6

30.9

99.3

59.1

2.4

0.5

1.9

25.4

20.0

06

0.0

35

1.5

3E

-06

25.4

425.4

225.4

20.0

05

0.0

32

1.3

9E

-06

64.1

30.7

99.0

43.0

2.5

0.4

2.0

25.4

00.0

07

0.0

38

1.8

1E

-06

25.4

025.4

025.3

90.0

06

0.0

34

2.1

1E

-06

60.7

25.8

94.8

64.3

2.5

1.1

1.5

25.2

20.0

13

0.0

69

6.2

8E

-06

25.2

425.2

225.2

20.0

12

0.0

65

4.5

8E

-06

60.2

26.7

94.7

39.5

3.1

0.9

2.2

25.2

00.0

14

0.0

75

7.5

0E

-06

25.1

925.2

025.2

00.0

12

0.0

69

9.9

1E

-06

53.8

20.2

88.6

63.9

3.8

1.2

2.6

24.9

10.0

44

0.2

27

1.0

1E

-04

24.9

624.9

124.9

10.0

37

0.2

10

5.9

3E

-05

50.5

16.1

85.4

41.9

3.8

1.3

2.5

24.7

70.0

76

0.3

90

3.8

9E

-04

24.7

724.7

724.7

70.0

66

0.3

62

3.3

2E

-04

49.9

15.5

82.6

51.8

5.4

2.8

2.5

24.7

40.0

85

0.4

33

5.0

4E

-04

24.7

424.7

424.7

40.0

73

0.4

02

3.9

9E

-04

49.3

17.3

83.6

40.7

5.7

1.7

4.1

24.7

10.0

92

0.4

71

6.2

1E

-04

24.7

324.7

124.7

10.0

73

0.4

18

4.8

5E

-04

49.4

16.6

82.3

45.7

6.3

2.7

3.6

24.7

20.0

91

0.4

66

6.0

5E

-04

24.7

224.7

224.7

20.0

74

0.4

19

4.8

4E

-04

49.2

17.1

82.5

36.0

6.5

2.5

4.1

24.7

10.0

93

0.4

77

6.4

2E

-04

24.7

024.7

124.7

10.0

74

0.4

23

6.3

1E

-04

49.0

16.8

79.6

44.6

8.8

4.8

4.1

24.7

00.0

97

0.4

96

7.0

7E

-04

24.6

724.7

024.7

00.0

77

0.4

40

6.3

1E

-04

49.7

16.5

79.3

35.9

9.2

5.9

3.3

24.7

30.0

87

0.4

45

5.4

0E

-04

24.6

424.7

324.7

30.0

72

0.4

04

8.8

4E

-04

49.3

15.1

77.0

49.0

11.0

8.3

2.7

24.7

10.0

92

0.4

73

6.2

8E

-04

24.6

324.7

124.7

10.0

79

0.4

38

6.5

4E

-04

50.5

16.8

77.7

40.2

11.8

8.9

2.9

24.7

70.0

76

0.3

88

3.8

5E

-04

24.6

324.7

724.7

70.0

64

0.3

58

7.2

6E

-04

50.1

18.2

77.5

38.4

12.8

8.6

4.1

24.7

50.0

81

0.4

17

4.6

0E

-04

24.6

224.7

524.7

50.0

64

0.3

70

8.3

6E

-04

49.0

18.0

77.1

35.2

12.9

7.9

5.0

24.7

00.0

96

0.4

92

6.9

3E

-04

24.5

925.4

224.7

00.0

73

0.4

26

1.2

2E

-03

49.4

16.6

74.9

40.0

14.8

11.3

3.5

24.7

20.0

91

0.4

66

6.0

6E

-04

24.5

524.7

224.7

20.0

75

0.4

20

1.0

7E

-03

50.1

15.4

73.8

36.2

16.5

14.3

2.2

24.7

50.0

81

0.4

15

4.5

3E

-04

24.4

924.7

524.7

50.0

72

0.3

87

1.8

5E

-03

49.5

17.1

69.2

44.0

24.4

20.5

3.9

24.7

20.0

90

0.4

58

5.8

0E

-04

24.4

524.7

224.7

20.0

72

0.4

08

1.1

6E

-03

50.6

16.7

68.2

44.2

26.6

23.8

2.8

24.7

70.0

75

0.3

87

3.8

1E

-04

24.4

324.7

724.7

70.0

65

0.3

58

1.1

9E

-03

50.1

15.6

67.4

45.3

26.7

24.3

2.4

24.7

50.0

81

0.4

14

4.5

1E

-04

24.4

224.7

524.7

50.0

71

0.3

85

1.1

8E

-03

49.8

20.9

69.8

36.3

27.1

20.1

7.0

24.7

40.0

85

0.4

34

5.0

9E

-04

24.4

225.4

724.7

40.0

58

0.3

57

2.1

9E

-03

50.2

18.0

68.2

39.0

27.1

23.1

4.0

24.7

50.0

81

0.4

13

4.4

9E

-04

24.3

924.7

524.7

50.0

65

0.3

67

2.1

9E

-03

50.3

18.0

67.9

38.5

27.8

24.0

3.8

24.7

60.0

78

0.4

02

4.1

8E

-04

24.3

824.7

624.7

60.0

64

0.3

60

2.4

6E

-03

49.0

16.6

64.5

39.7

31.1

27.1

3.9

24.7

00.0

98

0.4

98

7.1

5E

-04

24.3

024.7

024.7

00.0

79

0.4

43

3.0

9E

-03

50.5

16.8

62.5

38.0

35.4

32.5

2.9

24.7

60.0

76

0.3

92

3.9

4E

-04

24.2

624.7

624.7

60.0

68

0.3

61

5.3

4E

-03

49.7

19.0

60.4

36.7

39.7

34.5

5.2

24.7

30.0

86

0.4

41

5.2

9E

-04

24.2

225.4

924.7

30.0

70

0.3

80

6.9

1E

-03

50.4

15.5

57.3

36.8

42.5

40.3

2.2

24.7

60.0

78

0.4

00

4.1

4E

-04

24.1

524.7

624.7

60.0

78

0.3

74

1.3

5E

-02

49.9

15.4

56.8

36.9

42.8

40.4

2.4

24.7

40.0

83

0.4

27

4.8

6E

-04

24.1

424.7

424.7

40.0

81

0.3

97

1.3

5E

-02

50.1

16.3

52.2

38.9

50.3

47.4

2.9

24.7

50.0

82

0.4

19

4.6

5E

-04

24.1

324.7

524.7

50.0

77

0.3

86

1.4

7E

-02

50.6

19.7

53.9

35.7

50.6

45.6

5.0

24.7

70.0

76

0.3

87

3.8

2E

-04

24.1

225.4

424.7

70.0

71

0.3

36

1.9

7E

-02

50.3

18.7

53.0

35.1

51.0

46.6

4.3

24.7

60.0

79

0.4

03

4.2

2E

-04

24.0

924.7

624.7

60.0

78

0.3

56

2.4

4E

-02

12 of 12


Documents

Accounting for Future Redesign in the Optimization …...redesign small is obvious in that redesign can lead to project delays and increase costs. On the other hand, a reduction in